Numerical Evaluation of SDPA (Semidefinite Programming Algorithm)

  • Katsuki Fujisawa
  • Mituhiro Fukuda
  • Masakazu Kojima
  • Kazuhide Nakata
Part of the Applied Optimization book series (APOP, volume 33)


SDPA (SemiDefmite Programming Algorithm) is a C++ implementation of a Mehrotra-type primal-dual predictor-corrector interior-point method for solving the standard form semidefinite program and its dual. We report numerical results of large scale problems to evaluate its performance, and investigate how major time-consuming parts of SDPA vary with the problem size, the number of constraints and the sparsity of data matrices.


Search Direction Conjugate Gradient Method Semidefinite Program Minimum Eigenvalue Maximum Clique Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Alizadeh, F., J.-P.A. Haeberly and M.L. Overton. “Primal-dual interior-point methods for semidefinite programming,” Working Paper, 1994.Google Scholar
  2. [2]
    Alizadeh, F., J.-P.A. Haeberly and M.L. Overton. “Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results,” SIAM Journal on Optimization 8 (1998) 746–768.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Alizadeh, F., J.-P.A. Haeberly, M.V. Nayakkankuppam, M.L. Overton and S. Schmieta. “SDPpack — User’s Guide -,” Comp. Sci. Dept., New York University, New York, June 1997. Available at Scholar
  4. [4]
    Benson, S.J., Y. Ye and X. Zhang. “Solving large-scale sparse semidefinite programs for combinatorial optimization,” Applied Mathematics and Computer Sciences, The University of Iowa, Iowa City, Iowa 52242, September 1997.Google Scholar
  5. [5]
    Borchers, B. “CSDP, a C library for semidefinite programming,” Department of Mathematics, New Mexico Institute of Mining and Technology, 801 Leroy Place Socorro, New Mexico 87801, March 1997. Available at Scholar
  6. [6]
    Fujisawa, K., M. Kojima and K. Nakata.“SDPA (Semidefinite Programming Algorithm) — User’s Manual -,” Technical Report B-308, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 152, Japan, December 1995, Revised August 1996. Available at Scholar
  7. [7]
    Fujisawa, K., M. Kojima and K. Nakata. “Exploiting sparsity in primal-dual interior-point methods for semidefinite programming,” Mathematical Programming 79 (1997) 235–253.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Gahinet, P., and A. Nemirovski. “The projective method for solving linear matrix inequalities.”, Mathematical Programming 77 (1997) 163–190.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Golub, G.H., and C.F. Van Loan. Matrix Computations (second edition), (The John Hopkins University Press, Baltimore, Maryland, 1989).Google Scholar
  10. [10]
    Helmberg, C., and F. Rendl. “Solving quadratic (0, 1)-problems by semidefinite programming and cutting planes,” Mathematical Programming 82 (1998) 291–315.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Helmberg, C., F. Rendl, R.J. Vanderbei and H. Wolkowicz. “An interior-point method for semidefinite programming,” SIAM Journal on Optimization 6 (1996) 342–361.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Karisch, S.E., F. Rendl and J. Clausen. “Solving graph bisection problems with semidefinite programming,” to appear in INFORMS Journal on Computing.Google Scholar
  13. [13]
    Kojima, M., M. Shida and S. Shindoh. “Search directions in the SDP and the monotone SDLCP: generalization and inexact computation,” to appear in Mathematical Programming.Google Scholar
  14. [14]
    Kojima, M., S. Shindoh and S. Hara. “Interior-point methods for the monotone semidefinite linear complementarity problems,” SIAM. Journal on Optimization 7 (1997) 86–125.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Mehrotra, S. “On the implementation of a primal-dual interior point method,” SIAM Journal on Optimization 2 (1992) 575–601.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Lin, C.-J., and R. Saigal. “Semidefinite Programming and the Quadratic Assignment Problem,” 16th International Symposium on Mathematical Programming, Lausanne, Switzerland, August 1997.Google Scholar
  17. [17]
    Monteiro, R.D.C. “Primal-dual path-following algorithms for semidefinite programming,” SIAM Journal on Optimization 7 (1997) 663–678.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Monteiro, R.D.C., P.R. Zanjácomo. “Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya directions and their variants,” Technical Report, School Industrial and Systems Engineering, Georgia Tech., Atlanta, GA 30332, July 1997, Revised August 1997.Google Scholar
  19. [19]
    Nesterov, Yu.E., and M.J. Todd. “Self-scaled barriers and interior-point methods in convex programming,” Mathematics of Operations Research 22 (1997) 1–42.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Nesterov, Yu.E., and M.J. Todd. “Primal-dual interior-point methods for self-scaled cones,” SIAM J. on Optimization 8 (1988) 324–364.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Potra, F.A., R Sheng and N. Brixius. “SDPHA — a MATLAB im-plementation of homogeneous interior-point algorithms for semidefinite programming,” Department of Mathematics, University of Iowa, Iowa City, IA 52242, April 1997. Available at Scholar
  22. [22]
    Todd, M.J., K.C. Toh and R.H. Tütüncü. “On the Nesterov-Todd direction in semidefinite programming,” Technical Report, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853–3801, USA, March 1996, Revised May 1996, to appear in SIAM Journal on Optimization.Google Scholar
  23. [23]
    Toh, K.C., M.J. Todd and R.H. Tütüncü. “SDPT3 — a MATLAB software package for semidefinite programming,” Dept of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore, December 1996. Available at Scholar
  24. [24]
    Toh, K.C., and L.N. Trefethen. “The Chebyshev polynomial of matrix”, SIAM Journal on Matrix Analysis and Applications 20 (1998) 400–491.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Vandenberghe, L., and S. Boyd. “Semidefinite programming,” SIAM Review 38 (1996) 49–95.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Vandenberghe, L., and S. Boyd. “A primal-dual potential reduction method for problems involving matrix inequalities,” Mathematical Programming 69 (1995) 205–236.MathSciNetzbMATHGoogle Scholar
  27. [27]
    Wolkowicz, H., and Q. Zhao. “Semidefinite programming relaxations for the graph partitioning problem”, CORR Report, University of Waterloo Ontario, Canada, Oct. 1996.Google Scholar
  28. [28]
    Wu, S.-R., and S. Boyd. “SDPSOL: a parser/solver for SDP and MAXDET problems with matrix structure”, Department of Electrical Engineering, Stanford University, June, 1996. Available at Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Katsuki Fujisawa
  • Mituhiro Fukuda
  • Masakazu Kojima
  • Kazuhide Nakata

There are no affiliations available

Personalised recommendations